H S Hall and S R Knight Solutions for Chapter: Permutations and Combinations, Exercise 1: EXAMPLES. XI. a.

A boat is to be manned by eight men, of whom $2$ can only row on bow side and one can only row on stroke side; in how many ways can the crew be arranged?

There are two works, each of three volumes and two works each of two volumes, in how many ways can the books be placed on a shelf, so that the volumes of the same work are not separated?

In how many ways can $10$ examination papers be arranged so that the best and the worst papers never come together?

An eight-oared boat is to be manned by a crew chosen from $11$ men, of whom $3$ can steer but cannot row, and the rest can row but can not steer. In how many ways can the crew be arranged, if two of the man only row on the bow side?

Prove that the number of ways in which $p$ positive and $n$ negative signs may be placed in a row so that no two negative signs shall be together is ${}^{p+1}{C}_n$.

If ${}^{56}P_{r+6}{:}^{54}{P}_{r+3}=30800 :1$, find $r$.

How many different signals can be made by hoisting $6$ differently colored flags one above the other, when any number of them may be hoisted at once?

If ${}^{28}C_{2r} : {}^{24}C_{2r-4}=225 :11,$ find $r$.